The limit set of iterations of entire functions on wandering domains

TL;DR

This paper explores the types of domains that can appear as limit sets or invariant components in the dynamics of entire functions, revealing new possibilities for wandering and periodic domains.

Contribution

It demonstrates that any continuum without interior can be a limit set of an entire function on a wandering domain, and constructs entire functions with invariant Fatou components approaching a regular domain.

Findings

Any continuum without interior can be a limit set of an entire function on a wandering domain.

Constructed entire functions with invariant Fatou components approaching a regular domain.

Expanded understanding of possible domain types in complex dynamics.

Abstract

We first establish any continuum without interiors can be a limit set of iterations of an entire function on an oscillating wandering domain, and hence arise as a component of Julia sets. Recently, Luka Boc Thaler showed that every bounded connected regular open set, whose closure has a connected complement, is an oscillating or an escaping wandering domain of some entire function. A natural question is: What kind of domains can be realized as a periodic domain of some entire function? In this paper, we construct a sequence of entire functions whose invariant Fatou components can be approached to a regular domain.